Published online by Cambridge University Press: 12 March 2014
The purpose of this paper is to put on record some theorems relating to improvements in the primitive frame of combinatory logic. These improvements were, for the most part, suggested by the work of Rosser, who formulated a weakened system of combinatory logic in which the rules had a simple character not possessed by those of the original system. In the latter the rules B, C, W, K were in reality axiom-schemes, and their postulation amounted to assuming infinitely many axioms. Rosser had rules of procedure such that no propositions were deducible from them except in combination with axioms (or previously proved propositions); moreover, the conclusion of each rule was uniquely determined by the premises. He also eliminated equality as a primitive term, defining it (essentially) according to the traditional method. This paper shows that these and related advantages apply to certain formulations of the full system of combinatory logic, so far as it concerns the theory of combinators.
The method of procedure is as follows. Instead of setting up a primitive frame at the start and then deriving its properties, I begin (after some preliminary explanations in §2) by stating in §3 the properties which it is desired to establish. The next few sections are devoted to the formulation and proof of certain general theorems concerning possible bases for the system of §3. These theorems are, perhaps, more general than is necessary for the immediate purpose, but they are of some interest on their own account. A formulation in terms of the primitives of original system (i.e., B, C, W, and K), which is of the same general type as Rosser's formulation, is obtained at the end of §6. In §7 are discussed the changes in this formulation which are sufficient in order to base it on the primitive combinators S and K of Schönfinkel.
2 Cf. his thesis, A mathematical logic without variables, Annals of mathematics, vol. 36 (1935), pp. 127–150CrossRefGoogle Scholar, and Duke mathematical journal, vol. 1 (1935), pp. 328–355CrossRefGoogle Scholar. The introductory portions of Rosser's thesis, as they appear on a manuscript copy in my possession, did not appear when it was printed. Cf. also a footnote of Church in his paper, The Richard paradox, The American mathematical monthly, vol. 41 (1934), pp. 356–361CrossRefGoogle Scholar.
3 This paper is not yet published. However, a less elaborate formulation is given in §2 of my paper, The paradox of Kleene and Rosser, forthcoming in the Transactions of the American Mathematical Society. See also my address, Some aspects of the problem of mathematical rigor, forthcoming in the Bulletin of the American Mathematical Society.
4 For the present purposes the foregoing naive explanation of these connectives is probably sufficient. A more exact analysis of these connectives I have undertaken in a paper, Some properties of formal deducibility (not yet published). For those who wish a more accurate statement, it should be said that these connectives are used in the sense of that paper, which means that they are manipulated according to the rules of the Heyting Calculus (as formulated by Gentzen). In particular, → represents formal deducibility; this is not the most general sort of implication which can hold between two metatheoretic propositions (cf. the footnote to 2.42 of The paradox of Kleene and Rosser).
5 If ‘→’ is precisely defined, as indicated in footnote 4, the relation x⊦y defined in the text is equivalent to the following: if ⊦x is added to the axioms, then ⊦y follows by the axioms and rules of procedure, i.e., it is a theorem in the enlarged system. This equivalence, of course, is not obvious; its proof, as given in the paper cited in footnote 4, involves the Gentzen Hauptsatz, and is subject to restrictions on the formal system which are here automatically fulfilled. For the present purpose it is sufficient to take ‘→’ somewhat vaguely, and to note that if x⊦y should be defined as the deducibility relation just discussed, then x⊦y according to the definition in the text.
This use of ‘⊦’ as a relational sign is due to Rosser (loc. cit.), who defined it slightly differently.
6 This is not quite the same as ∣R∣X. Cf. Lemma 4, below.
7 In the following theorems it is not necessary that the variables used in formulating properties of Q should range over all terms. In fact the theorems all hold if these variables range over any subset of these terms which includes B, C, W, K and is closed under application. The variables in connection with the properties of R and its specialization ⊦(including the properties given as rules for Ξ and E) range over the more extended class. However, in the rules R∣[K]∣, which play no essential role in the argument, they may instead be restricted to range over the same subclass as for the rules relating to Q. This rather slight generalization is made use of in the statement of Theorem 17. Further generalizations are, of course, possible—cf. Remark 2 to Theorem 17—but such generalizations are not explicitly formulated here.
8 R(c) is used only to prove (2). In GKL, (2) follows from a special form of the reflexive law of identity.
9 The hypotheses of Theorem 13, for R interpreted as [⊦], hold by the hypothesis of this theorem and Theorem 9.
10 An alternative procedure would be to define Q by Q = ΨΞ T, where Ψ is the combinator defined in my paper, The universal quantifier in combinatory logic, Annals of mathematics, vol. 32 (1931), pp. 154–180CrossRefGoogle Scholar, see §4, Definition 2 (this paper will hereafter be referred to as ‘UQ’). The property Ψuvxy R u(vx)(vy) is then a consequence of R3rBCW; hence the proof of the theorem goes through if [R]W is added to the hypotheses. Theorems 16, 17, and 25 would also be valid.
11 The theorem is also valid if we define T as CI. Then we have to use [R]3 in the derivation of (4) and (5).
12 At the second application [R]B alone suffices; at the first application [R]3 has to be applied twice.
13 An alternative scheme is to define E as WQ and assume the above four axioms and rule.
14 It is to be understood in the case of the rule [⊦](K) that it is stated in the form ⊦ux & ⊦Ey → ⊦u(Kxy). This is so that Rosser's criterion for a rule of procedure be satisfied.
15 There are two remarks, besides those in the text, which may be added at this point. The first is that, if additional axioms of the form ⊦EX are added to the primitive frame, then Q is an adequate equality relation for the first system of §3 with these additional terms adjoined to I (so that the terms of the derived system consist of all the terms of the underlying system for which ⊦EX holds).
The second remark arises from the universal character of E. It is evident that propositions of the form ⊦EX are trivial; if we were not interested in making the rules have the Rosser form, such propositions might be dropped into the morphology (cf. the second paper cited in footnote 3). But if these are formulated here then metatheorems involving variables are subject to the restriction that ⊦Ex holds for each variable x appearing in the theorem.
16 The full rule Ξ is not derivable from these rules (unless ⊦E Ξ and Ξ EE are added as new axioms), but only Rule Ξ with the variables ranging over the more restricted class of terms. However this suffices for the proof of Theorem 17, since the weaker form of the rule is sufficient for Theorem 15, while the new axiom which replaces Axiom Q 0 eliminated the dependence on Rule Ξ in Theorem 16.
17 Axiom Q 0 is, of course, to be retained.
18 If we make the same reservation in regard to [Ξ](K) as in Theorem 17, then it is evident, since this rule is involved essentially in Theorem 23, even for properties expressible in terms of B, C, W, without K, that we do not have the same simple situation mentioned in footnote 7. In this case we add to III A the axiom ΞEΞ. Whether or not the theorem will go through without this extra hypothesis is not here investigated.
The remarks of footnote 15 also apply here.